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Bingyao Wu

Publications and source records attributed to Bingyao Wu.

11 recordsLinked to original sources
arXiv2025

Convergence rates for the $p$-Wasserstein distance of the empirical measures of an ergodic Markov process

Let $X:=(X_t)_{t\geq 0}$ be an ergodic Markov process on $\real^d$, and $p>0$. We derive upper bounds of the $p$-Wasserstein distance between the invariant measure and the empirical measures of the Markov process $X$. For this we assume, e.g.\ that the transition semigroup of $X$ is exponentially contractive in terms of the $1$-Wasserstein distance, or that the iterated Poincaré inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.

math.PR↗
arXiv2025

Exact rate of convergence for the empirical measure of a subordinated process in $p$-Wasserstein distance

We establish exact rates of convergence in the $p$-Wasserstein distance for the empirical measure of a class of non-symmetric jump processes, which are subordinated to a diffusion process on a compact Riemannian manifold. For the quadratic Wasserstein distance, we determine the renormalization limit. We extend the main results of \cite{WW} and \cite{WWZ}. Our method uses two key elements: a Bernstein-type inequality for the subordinated process and the PDE approach established in \cite{AMB}

math.PR↗
arXiv2025

On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting

We investigate the limiting behavior of Besov seminorms and nonlocal perimeters in Dunkl theory. The present work generalizes two fundamental results: the Maz'ya--Shaposhnikova formula for Gagliardo seminorms and the asymptotics of (relative) fractional $s$-perimeters. Our main contributions are twofold. First, we establish a dimension-free Maz'ya--Shaposhnikova formula via a novel, robust approach that avoids reliance on the density property of Besov spaces, offering broader applicability. Second, we prove limit formulas for nonlocal perimeters relative to bounded open sets $Ω$, removing boundary regularity assumptions in the forward direction, while introducing a weakened regularity condition on $\partialΩ$ (admitting fractal boundaries) for the converse, a significant improvement over existing requirements. To the best of our knowledge, the results in this second part are new even in the classic Laplacian setting.

math.FA↗
arXiv2024

Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{nα\}$

Fix an irrational number $α$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $φ$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random walk $\{S_n α\}_{n\ge 1}$ on the torus, where $\{\cdot\}$ denotes the fractional part. We study the long time asymptotic behaviour of the empirical measure of this random walk to the uniform distribution under the general $p$-Wasserstein distance. Our results show that the Wasserstein convergence rate depends on the Diophantine properties of $α$ and the Hölder continuity of the characteristic function $φ$ at the origin, and there is an interesting critical phenomenon that will occur. The proof is based on the PDE approach developed by L. Ambrosio, F. Stra and D. Trevisan in [2] and the continued fraction representation of the irrational number $α$.

math.PR↗
arXiv2024

Sharp $L^q$-Convergence Rate in $p$-Wasserstein Distance for Empirical Measures of Diffusion Processes

For a class of (non-symmetric) diffusion processes on a length space, which in particular include the (reflecting) diffusion processes on a connected compact Riemannian manifold, the exact convergence rate is derived for $({\mathbb E} [{\mathbb W}_p^q(μ_T,μ)])^{\frac{1}{q}} (T \to \infty)$ uniformly in $(p,q)\in [1,\infty) \times (0,\infty)$, where $μ_T$ is the empirical measure of the diffusion process, $μ$ is the unique invariant probability measure, and ${\mathbb W}_p$ is the $p$-Wasserstein distance. Moreover, when the dimension parameter is less than $4$, we prove that ${\mathbb E} |T {\mathbb W}_2^2(μ_T,μ)-Ξ(T)|^q \to 0$ as $T\to\infty$ for any $q\ge 1$, where $Ξ(T)$ is explicitly given by eigenvalues and eigenfunctions for the symmetric part of the generator.

math.PR↗
arXiv2023

Asymptotics of Kantorovich Distance for Empirical Measures of the Laguerre Model

We estimate the rate of convergence for the Kantorovich (or Wasserstein) distance between empirical measures of i.i.d. random variables associated with the Laguerre model of order $α$ on $(0,\infty)^N$ and their common law, which is not compactly supported and has no rotational symmetry. Compared with the Gaussian case, our result is sharp provided the parameter $α$ and the dimension $N$ are chosen in a specified regime.

math.PR↗
arXiv2023

Wasserstein Convergence for Empirical Measures of Subordinated Fractional Brownian Motions on the Flat Torus

We estimate rates of convergence for empirical measures associated with the subordinated fractional Brownian motion to the uniform distribution on the flat torus under the Wasserstein distance $\mathbb{W}_p$ for all $p\geq1$. In particular, our results coincides with recent ones on the diffusion process and the fractional Brownian motion. As an application, we provide similar results for time-discretized subordinated fractional Brownian motions.

math.PR↗
arXiv2022

Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds

The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than [arXiv:2107.11568], generalize the recent ones in [Stoch. Proc. Appl. 144(2022), 271--287] and are shown to be sharp by a typical example. The proof is motivated by the aforementioned works.

math.PR↗
arXiv2022

Wasserstein Convergence for Empirical Measures of Subordinated Dirichlet Diffusions on Riemannian Manifolds

We investigate long-time behaviors of empirical measures associated with subordinated Dirichlet diffusion processes on a compact Riemannian manifold $M$ with boundary $\partial M$ to some reference measure, under the quadratic Wasserstein distance. For any initial distribution not concentrated on $\partial M$, we obtain the rate of convergence and even the precise limit for the conditional expectation of the quadratic Wasserstein distance conditioned on the process killed upon exiting $M\setminus\partial M$. In particular, the results coincide with the recent ones proved by F.-Y. Wang in \cite{eW2} for Dirichlet diffusion processes.

math.PR↗
arXiv2022

Wasserstein Convergence for Conditional Empirical Measures of Subordinated Dirichlet Diffusions on Riemannian Manifolds

The asymptotic behaviour of empirical measures has plenty of studies. However, the research on conditional empirical measures is limited. Being the development of Wang \cite{eW1}, under the quadratic Wasserstein distance, we investigate the rate of convergence of conditional empirical measures associated to subordinated Dirichlet diffusion processes on a connected compact Riemannian manifold with absorbing boundary. We give the sharp rate of convergence for any initial distribution and prove the precise limit for a large class of initial distributions. We follow the basic idea of Wang, but allow ourselves substantial deviations in the proof to overcome difficulties in our non-local setting.

math.PR↗